Nasty behaviour of sqrt(x)

[dpsanders]

I think the following should throw an error instead.

julia> x, y = set_variables("x y", order=8)
2-element Array{TaylorSeries.TaylorN{Float64},1}:
  1.0 x + 𝒪(‖x‖⁹)
  1.0 y + 𝒪(‖x‖⁹)

julia> sqrt(x)
 Inf x - NaN y - Inf x² - NaN x y - NaN y² + Inf x³ - NaN x² y - NaN x y² - NaN y³ - Inf x⁴ - NaN x³ y - NaN x² y² - NaN x y³ - NaN y⁴ + Inf x⁵ - NaN x⁴ y - NaN x³ y² - NaN x² y³ - NaN x y⁴ - NaN y⁵ - Inf x⁶ - NaN x⁵ y - NaN x⁴ y² - NaN x³ y³ - NaN x² y⁴ - NaN x y⁵ - NaN y⁶ + Inf x⁷ - NaN x⁶ y - NaN x⁵ y² - NaN x⁴ y³ - NaN x³ y⁴ - NaN x² y⁵ - NaN x y⁶ - NaN y⁷ - Inf x⁸ - NaN x⁷ y - NaN x⁶ y² - NaN x⁵ y³ - NaN x⁴ y⁴ - NaN x³ y⁵ - NaN x² y⁶ - NaN x y⁷ - NaN y⁸ + 𝒪(‖x‖⁹)

[dpsanders]

I guess when the constant term is zero.

[dpsanders]

Same for log

[dpsanders]

I now get

x, y = set_variables("x y", order=8)
julia> sqrt(x^2)
ERROR: ArgumentError("The 0-th order TaylorN coefficient must be non-zero\nin order to expand `sqrt` around 0")
 in sqrt at /Users/dsanders/.julia/v0.3/TaylorSeries/src/TaylorN.jl:615

but it works for one variable:

julia> t = taylor1_variable(10)
 1.0 t + 𝒪(t¹¹)

julia> sqrt(t^2)
 1.0 t + 𝒪(t¹¹)

[lbenet]

Yes, you are right; this is something known.

In simple terms, Taylor1 objects are simpler to handle and we know the way to factorize them, or to recognize that sqrt(t^2) is t. The situation is different for TaylorN objects (under the current representation we are using!). I haven't had time to think about this, but indeed this is something to be solved.

This problem does not appear if we construct TaylorN objects as Taylor1 objects, whose coefficients are polynomials in other variables. This is part of the idea in branch "TaylorRec"; see #12.

[lbenet]

In my opinion, this issue should be closed, but an issue reminding us about this should definitely be open.

[lbenet]

See also discussion in #11